Modular arithmetic of mancala games
Modular arithmetic of mancala games → Portuguese. The connection between mancala games and modular arithmetic (also known as clock arithmetic) is that, in most games, seeds are distributed along a closed path (ie. a path with the same first and last vertex). Modular arithmetic is most useful, when it is important to know where the last seed is sown. It works best in single-lap games. Examples Each pit is represented by an integer and numbers "wrap around" after the sowing reaches the emptied pit again. The modulus n'' is defined by the number of pits that form the closed path of distribution. If, for instance, the game is played on a 2 x 4 board and the original pit isn't skipped, the modulus is 8 and 3 is congruent to 11, 19, 27 and so on, because the remainder is for each dividend 3 after it was divided by 8. The quotient gives the number of cycles. '''The following formulae are used to determine the pit, which receives the last seed of a sowing:' Vai Lung Thlan * Board: 2 rows x 6 pits. * First seed is sown in the next pit. * Emptied hole isn't skipped. : (e + s mod12) mod12 = d Oware * Board: 2 rows x 6 pits. * First seed is sown in the next pit. * Emptied hole is skipped. : If s'' < 12: : ''(e + s mod12) mod12 = d : If s'' > 11: : ''(e + (s + 1) mod12) mod12 = d Cross-Wari * Board: 2 rows x 6 pits. * First seed is sown in the next pit. * Emptied hole is skipped. : If s'' < 12 and even: : ''(e + s mod12) mod12 = d : If s'' < 12 and odd: : ''(e-s mod12) mod12 = d : If s'' > 11 and even: : ''(e + (s + 1) mod12) mod12 = d : If s'' > 11 and odd: : ''(e - (s + 1) mod12) mod12 = d Toguz Kumalak * Board: 2 rows x 9 pits. * First seed is sown in the emptied pit unless it is a singleton, which is dropped into the next pit. * Emptied hole isn't skipped. : If s'' = 1: : ''e + 1 = d : If s'' > 1: : ''(e + (s-1) mod18) mod18 = d The following formula is used to determine how many seeds a pit must contain to reach the player's store with the last seed: Kalah * Board: 2 rows x 6 pits + 1 functional endhole for each player at his right. * First seed is sown in the next pit (or the player's store). * Emptied hole isn't skipped. : s = (7 - e) mod13 Variables e'': emptied pit ''d: destination pit s'': seeds in the emptied pit See also * cycles External Links * Modular arithmetic Copyright ''© Ralf Gering Under the CC by-sa 2.5 license. Category:Mathematics